In this paper we consider the linear, time-dependent quantum Harmonic Schrdinger equation ${\textrm {i}} \partial _t u= \frac {1}{2} ( - \partial _x^2 + x^2) u + V(t, x, D)u$, $x \in \mathbb {R}$, where $V(t,x,D)$ is classical pseudodifferential operator of order 0, self-adjoint, and $2\pi $ periodic in time. We give sufficient conditions on the principal symbol of $V(t,x,D)$ ensuring the existence of solutions displaying infinite time growth of Sobolev norms. These conditions are generic in the Fréchet space of symbols. This shows that generic, classical pseudodifferential, $2\pi $-periodic perturbations provoke unstable dynamics. The proof builds on the results of [36] and it is based on pseudodifferential normal form and local energy decay estimates. These last are proved exploiting Mourre’s positive commutator theory.

中文翻译：

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ℝ 上线性时间相关量子谐波振荡器的通用传输器

在本文中，我们考虑线性、时间相关的量子谐波薛定谔方程 ${\textrm {i}} \partial _t u= \frac {1}{2} ( - \partial _x^2 + x^2) u + V(t, x, D)u$, $x \in \mathbb {R}$，其中 $V(t,x,D)$ 是经典的 0 阶伪微分算子，自伴，$2\pi $在时间上周期性。我们给出了主符号 $V(t,x,D)$ 的充分条件，以确保存在显示 Sobolev 范数无限时间增长的解。这些条件在符号的 Fréchet 空间中是通用的。这表明一般的、经典的伪微分、$2\pi $-周期性扰动会引发不稳定的动力学。该证明建立在 [36] 的结果之上，它基于伪微分范式和局部能量衰减估计。最后这些被证明是利用 Mourre 的正交换器理论。